“There’s no question we have serious differences, and this is going to continue to be a difficult conversation,” Karine Jean-Pierre, the White House press secretary, told reporters in Japan. “That’s not lost on us. But the president’s team is going to continue to work hard toward a reasonable bipartisan solution that can pass the House and the Senate.”

In a nod to growing complaints on the left, Ms. Jean-Pierre emphasized the need for both Republican and Democratic votes. And when pressed by reporters on the more measured tone, she insisted that “the optimism continues to be there,” while adding several times that a deal would depend on whether Mr. McCarthy “will negotiate in good faith” and that everyone should recognize that “you don’t get everything that you want.”

Both Democratic and Republican leaders were facing pressure from their bases not to compromise.

Former President Donald J. Trump weighed in on Friday on his social media site, declaring that Republicans should not make a deal on the debt ceiling unless they got everything they wanted.

“DO NOT FOLD!!!” he wrote.

In a letter, liberal Democratic lawmakers renewed their calls for Mr. Biden to “refuse to reward Republicans’ reckless refusal to raise the debt ceiling without preconditions,” urging him instead to invoke the 14th Amendment to continue issuing new debt to pay bondholders, Social Security recipients, government employees and others.

Negotiators were at odds over a handful of issues, including the extent to which a possible deal would include tougher work requirements for social safety net programs — a proposal that has drawn a backlash from progressive Democrats — and the length of any debt limit extension.